What is the value of `(1)/( 3 xx 7) + (1)/( 7 xx 11) + (1)/( 11 xx 15) + . . . + (1)/(899 xx 903)` ?

Options

A. 21/509
B. 18/403
C. 18/403
D. 29/31

Answer

To solve the problem, we need to evaluate the sum:

\[ S = \frac{1}{3 \times 7} + \frac{1}{7 \times 11} + \frac{1}{11 \times 15} + \ldots + \frac{1}{899 \times 903} \]

### Step 1: Identify the pattern in the series

The general term of the series can be expressed as:

\[ \frac{1}{(4n - 1)(4n + 3)} \]

where \( n \) starts from 1 and goes up to 225 (since \( 4n - 1 = 899 \) when \( n = 225 \)).

### Step 2: Rewrite the general term using partial fractions

We can express the general term using partial fractions:

\[ \frac{1}{(4n - 1)(4n + 3)} = \frac{A}{4n - 1} + \frac{B}{4n + 3} \]

Multiplying through by the denominator:

\[ 1 = A(4n + 3) + B(4n - 1) \]

### Step 3: Solve for A and B

Expanding and equating coefficients, we have:

\[ 1 = (4A + 4B)n + (3A - B) \]

Setting the coefficients equal gives us the system:

1. \( 4A + 4B = 0 \)
2. \( 3A - B = 1 \)

From the first equation, \( A + B = 0 \) implies \( B = -A \). Substituting into the second equation:

\[ 3A - (-A) = 1 \implies 4A = 1 \implies A = \frac{1}{4} \]
\[ B = -\frac{1}{4} \]

### Step 4: Rewrite the general term

Thus, we can rewrite the general term as:

\[ \frac{1}{(4n - 1)(4n + 3)} = \frac{1/4}{4n - 1} - \frac{1/4}{4n + 3} \]

### Step 5: Write the sum

Now we can write the sum \( S \) as:

\[ S = \frac{1}{4} \left( \left( \frac{1}{3} - \frac{1}{7} \right) + \left( \frac{1}{7} - \frac{1}{11} \right) + \left( \frac{1}{11} - \frac{1}{15} \right) + \ldots + \left( \frac{1}{899} - \frac{1}{903} \right) \right) \]

### Step 6: Notice the telescoping nature

The series telescopes, meaning most terms cancel out:

\[ S = \frac{1}{4} \left( \frac{1}{3} - \frac{1}{903} \right) \]

### Step 7: Calculate the final value

Calculating the remaining terms:

\[ S = \frac{1}{4} \left( \frac{1}{3} - \frac{1}{903} \right) = \frac{1}{4} \left( \frac{903 - 3}{3 \times 903} \right) = \frac{1}{4} \left( \frac{900}{3 \times 903} \right) = \frac{1}{4} \left( \frac{300}{903} \right) \]

### Step 8: Simplify

Finally, simplifying gives:

\[ S = \frac{75}{903} = \frac{25}{301} \]

### Final Answer

Thus, the value of the sum is:

\[ \boxed{\frac{25}{301}} \]